2- or 4-piece sound system for $750 or less?

Is what your saying you need 4 speakers for $750??
You won't get 4 good speakers for that unless they are very small passive. Then you need a power amp which will cost that amount too.  For that price point you are not looking at the top brands at all. Example that's the price for just one Yamaha , QSC or JBL powered speaker. Have you not looked on the Sweetwater or MF site yet?? 
 
For sure you will need powered speakers. The best bang for the buck is Alto. Excellent sound and so far hard to find a negative review. The 10" are fine too if just for simple acoustic music.  I've had mine for more than a year now and I'm very pleased. Lots of clean power and even though they are only 12" they kick out a good low end. I'm not a fan of 15" speakers anyhow. 
https://www.altoproaudio.com/products/ts212
  
That's not a good idea to put speakers at the rear of the room unless you understand how to set them up with a delay, very advanced skills needed for that. If you can fly the speakers or just get them up there on stands. 
 Shopping used or demos in music stores is a great plan-I picked up a pair of Mackie 10" powered speakers that had been demos for only $150 each. New they were $450 each
 
Analog mixers are very affordable these days. My top pick last year was a Yamaha MPG 16. That handled a 5 piece band no problem. I chose it over the others because of the feature set and our needs. Every situation is different. Like how many monitors and Aux you need.
Make sure to count the XLR inputs and faders. They tend to call mixers 12 channels when It's really 8.  
This is the mixer I would buy this time because I'm in a 3 piece band now.   https://www.sweetwater.com/store/detail/Sig12MT 
But I will probably go with a Behringer X12 air. Which is worth looking into if your tech savvy and have an iPad.   
 
 

ampfixer
Your best bet at that price will be the used market. Hit the papers and look for guy that's dumping his PA system.

 
Like me, I just sold my Yamaha club series 12" with 2" horns for $200. Perfect condition. The guy who is in a duo I sometimes play bass with,  was happy as he had a good power amp already but was using these terrible Community's. Worst speaker I'd ever sang though. He also just bought a Behringer X32! Just him and his wife ? He paid over $6,000 for his Pro Tools set up. It's nice but?? 

"I'll be on the lookout for a good mixer and some used speakers. I know they're out there, but I haven't been keeping my eyes peeled for 'em! I don't need to exclusively be looking for PA speakers though, do I? Some studio monitors can meet the specs as well, I'm guessing?"
 
Yes you do need to look at PA speakers. Using anything else is compromising your sound.  Studio monitors are more like 100 Watts and your looking at 300 watts plus for a room like that. PA speakers are designed for what they do, studio monitors are designed for what they do. 
 
Use the Sweetwater and Musicians Friend sites to look at gear, Read the reviews as you go. That is why I ended up with Altos. A balance of price point and positive reviews form all. Once I narrow things down I always go and buy at a local music store. They always price match best they can too. 

Well the Altos are very portable. Weight was a big factor for me when I was comparing speakers. They have 3  handles too top and sides.   Birch plywood was common and what we used for the last 4 decades but now most boxes are moulded which not only brought down manufacturing prices but made them lighter. Even though my Alto's are powered they way a few pounds less than my old Yamaha unpowered boxes. That's why I would never go back to un powered systems. My power amp weighed 80Lbs!! 
 
Look at the Mackie, Yamaha, Soundcraft and Allen &Heath small mixers. Make sure there is 8 XLR's. And if your going to need stage monitors make sure you have at least 2 Aux sends plus effects.